Investigating the use of models to develop Grade 8 learners’ conceptual understanding of and procedural fluency with fractions

Both my teaching experience and literature of this research study strongly suggested that fractions are difficult to teach and learn across the globe generally, and Namibia in particular. One of the identified contributing factors was teaching fractions by focusing on procedures and not the conceptual understanding. Therefore, this research project developed and implemented an intervention in order to experiment and suggest an alternative teaching approach of fractions using models. The purpose of this research was to: “Investigate the use of models to develop Grade 8 learners’ conceptual understanding of and procedural fluency with fractions”. This investigation had three areas of focus. Firstly, the study investigated the nature of learners’ conceptual understanding of and procedural fluency with fractions before the teaching intervention, by means of administering a pre-test and pre-interview and analysing learners’ responses. Secondly, the study investigated the changes in learners’ conceptual understanding of and procedural fluency with fractions after the teaching intervention, by means of administering a post-test, post-interviews and recall interviews, and analysing learners’ responses. Thirdly, this study investigated the possible influence of the teaching intervention on the changes in learners’ conceptual understanding of and procedural fluency with fractions by analysing the lesson videos and learners’ worksheets, and describe their critical interaction. This study was conducted at a multicultural urban secondary school located in the Oshikoto Region, Namibia. The sample consisted of 12 Grade 8 mathematics learners whose age ranged from 13-16 years old. A purposive sampling method was employed to select both the research site and participants. This research is framed as a case study, and is grounded within the interpretive paradigm and qualitative research. This research revealed that these learners displayed conceptual and procedural difficulties in their engagement with fraction models and fraction symbols, before the teaching intervention. Conceptually, the study found that these learners read fractions using inappropriate names; and learners did not identify the whole unit in the models and therefore identified fractions represented by the fraction models using different forms of inappropriate fraction symbols. Procedurally, the study found that these learners compared and ordered fractions inappropriately using the sizes of the numerators and denominators separately; and learners used the lowest common denominator method inappropriately for adding fractions with different denominators. The research also suggested conceptual and procedural changes in learners’ conceptual understanding of and procedural fluency with fractions and that the intervention seemed to help learners to engage better with fraction models and fraction symbols. Conceptually, the findings suggested that the intervention using area models and number lines, seemed to help these learners to read fractions using appropriate names; to identify the whole unit in the fraction models and to develop a sense of the size of fractions in relation to one whole unit. Procedurally, the learners compared and ordered fractions appropriately using either equal fraction bars, equal number lines, benchmarking or rules for comparing and ordering fractions with the same numerator or denominator; and learners used equal fraction bars to visually represent the lowest common denominator method and to recognise that only equally sized units can be counted together. This research identified four factors as possible influences of the teaching intervention. These factors are namely: identifying both fraction symbols and appropriate fraction names to see fractions as relational numbers; prompting to partition whole units of the fraction models and graphically illustrating fraction symbols to identify the whole unit in the fraction models and to develop a sense of the size of fractions in relation to one whole unit; graphically illustrating fraction symbols using the models to use equal fraction bars and number lines, benchmarking and rules for comparing; and graphically illustrating fraction denominations using equal fraction bars to recognise that only equally sized units can be counted together. This research strongly suggests that the effective use of models has the potential to develop learners’ conceptual understanding of and procedural fluency with fractions in a number of ways